Substitutions for integrals involving branch cuts

Question

I have the definition of the Liouville fractional derivative \begin{equation} \frac{d^{-\delta}}{d x^{-\delta}} f(x) = \int_{0}^x f(t)(t-x)^{\delta - 1} dt \end{equation} where $\delta > 0$ and for our considerations is fractional. There is a branch point singularity at $t = x$ of the integrand. I want to find out the representation of the derivative operator \begin{equation} T^\delta_x = \frac{d^{-\delta}}{d\big(-\frac{1}{x}\big)^{-\delta}}f(x) \end{equation} I notice that the simple subsitutions $x\to -\frac{1}{x}$ and then $f(x) \to f\big(-\frac{1}{x}\big)$ do not work on their own as they don't give me the right answer which is \begin{equation} T^\delta_x f(x) = x^{1-\delta}\int_0^x f(t) (x-t)^{\delta -1}t^{-1-\delta} dt. \end{equation} I suspect that it is because I am not taking care of the branch cut but I am unable to resolve this problem on my own. Any help is greatly appreciated. Thank you.